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Magazine Chapter 5: Problem 38
Simplify each numerical expression. \(2^{-4}+5^{-1}\)
Short Answer
Expert verified
The simplified expression is \( \frac{21}{80} \).
Step by step solution
01
Simplify the First Term
Understand and simplify the first component of the expression, which is \(2^{-4}\). Recall that a negative exponent can be rewritten as a reciprocal of the positive exponent. Therefore, \(2^{-4} = \frac{1}{2^4}\). Calculate \(2^4 = 2 \times 2 \times 2 \times 2 = 16\), so \(2^{-4} = \frac{1}{16}\).
02
Simplify the Second Term
Now, simplify the second component, which is \(5^{-1}\). Similarly, negative exponents denote the reciprocal of the base raised to the corresponding positive power. Therefore, \(5^{-1} = \frac{1}{5^1} = \frac{1}{5}\).
03
Combine the Simplified Terms
Add the two simplified fractions from the previous steps. We have \(\frac{1}{16} + \frac{1}{5}\). To perform this addition, we need a common denominator. The least common multiple of 16 and 5 is 80.
04
Convert to a Common Denominator
Convert \(\frac{1}{16}\) and \(\frac{1}{5}\) to equivalent fractions with a common denominator of 80. Calculate \(\frac{1}{16} = \frac{5}{80}\) and \(\frac{1}{5} = \frac{16}{80}\). We do this by multiplying the numerator and denominator of \(\frac{1}{16}\) by 5, and for \(\frac{1}{5}\), by 16.
05
Add the Fractions
Now, add the fractions with the common denominator: \(\frac{5}{80} + \frac{16}{80} = \frac{5+16}{80} = \frac{21}{80}\).
06
Simplify if Possible
Check if \(\frac{21}{80}\) can be further simplified. Since 21 and 80 have no common factors other than 1, the fraction is already in its simplest form.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Negative Exponents
Negative exponents can seem confusing at first, but they have a straightforward rule to follow. When you see a negative exponent, it means you take the reciprocal of the base raised to the positive of that exponent. For example, with \(2^{-4}\), you switch from \(2\) raised to the negative fourth power to its reciprocal, which is \(\frac{1}{2^4}\).
Here's a simple step-by-step:
Here's a simple step-by-step:
- Identify the negative exponent, such as in \(5^{-1}\).
- Rewrite the expression using the reciprocal, \(5^{-1} = \frac{1}{5}\).
- Calculate the positive power of the base, like \(2^4 = 16\).
- Wrap it up to get, \(2^{-4} = \frac{1}{16}\).
Fractions
Fractions represent a part of a whole and are composed of a numerator over a denominator, like \(\frac{a}{b}\). In the context of simplifying expressions with negative exponents, fractions come into play when rewriting and adding terms.
When you simplify an expression with negative exponents, like \(2^{-4}\) and \(5^{-1}\), you convert these into fractions. Specifically, \(2^{-4}\) becomes \(\frac{1}{16}\), and \(5^{-1}\) becomes \(\frac{1}{5}\).
Understanding fractions deeply:
When you simplify an expression with negative exponents, like \(2^{-4}\) and \(5^{-1}\), you convert these into fractions. Specifically, \(2^{-4}\) becomes \(\frac{1}{16}\), and \(5^{-1}\) becomes \(\frac{1}{5}\).
Understanding fractions deeply:
- The **denominator** tells how many equal parts the whole is divided into.
- The **numerator** shows how many parts we are considering.
- Adding fractions requires a common denominator.
Least Common Denominator
The least common denominator (LCD) is a helpful concept when adding or subtracting fractions. It is the smallest shared multiple in the denominators of the fractions involved. When you combine fractions like \(\frac{1}{16}\) and \(\frac{1}{5}\), your goal is to make the denominators the same.
Here's how to find the least common denominator:
Here's how to find the least common denominator:
- List multiples of each denominator. For \(16\) and \(5\), those are:
- 16: 16, 32, 48, 64, 80, ...
- 5: 5, 10, 15, 20, 25, 30, 80, ...
- Identify the smallest common multiple, which here is \(80\).
- Convert each fraction to an equivalent fraction with this common denominator:
- \(\frac{1}{16} = \frac{5}{80}\)
- \(\frac{1}{5} = \frac{16}{80}\)
